Two-derivative error inhibiting schemes with post-processing

TL;DR

This paper extends error inhibiting techniques to two-derivative general linear methods, enabling higher accuracy and stability through post-processing and optimization of explicit and implicit schemes for solving differential equations.

Contribution

It introduces a theoretical framework for two-derivative GLMs with error control and develops optimized explicit and implicit methods up to eighth order with favorable stability properties.

Findings

Methods achieve higher order accuracy through post-processing.

Explicit SSP methods up to seventh order are constructed.

Numerical experiments confirm theoretical convergence and stability.

Abstract

High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semi-discretization of partial differential equations. In prior work in we investigated the interplay between the local truncation error and the global error to construct error inhibiting general linear methods (GLMs) that control the accumulation of the local truncation error over time. Furthermore we defined sufficient conditions that allow us to post-process the final solution and obtain a solution that is two orders of accuracy higher than expected from truncation error analysis alone. In this work we extend this theory to the class of two-derivative GLMs. We define sufficient conditions that control the growth of the error so that the solution is one order higher than expected from truncation error analysis, and furthermore define the construction of a simple post-processor that will extract an additional order of accuracy. Using these conditions as constraints, we develop an optimization code that enables us to find explicit two-derivative methods up to eighth order that have favorable stability regions, explicit strong stability preserving methods up to seventh order, and A-stable implicit methods up to fifth order. We numerically verify the order of convergence of a selection of these methods, and the total variation diminishing performance of some of the SSP methods. We confirm that the methods found perform as predicted by the theory developed herein.